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Similar to the examples described above, we consider x, y, φ to be independent uniform random variables over the ranges 0 ≤ x ≤ a, 0 ≤ y ≤ b, − π / 2 ≤ φ ≤ π / 2 . To solve such a problem, we first compute the probability that the needle crosses no lines, and then we take its complement.
We can convert 2D points to homogeneous coordinates by defining them as (x, y, 1). Assume that we want to find intersection of two infinite lines in 2-dimensional space, defined as a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0. We can represent these two lines in line coordinates as U 1 = (a 1, b 1, c 1) and U 2 = (a 2, b 2, c 2).
The analytic determination of the intersection curve of two surfaces is easy only in simple cases; for example: a) the intersection of two planes, b) plane section of a quadric (sphere, cylinder, cone, etc.), c) intersection of two quadrics in special cases. For the general case, literature provides algorithms, in order to calculate points of ...
As t goes from 0 to 1, the point follows the part of the circle in the first quadrant from (1, 0) to (0, 1). Finally, as t goes from 1 to +∞, the point follows the part of the circle in the second quadrant from (0, 1) to (−1, 0). Here is another geometric point of view. Draw the unit circle, and let P be the point (−1, 0).
The measure of ∠AOB, where O is the center of the circle, is 2α. The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that intercepts the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle.
In general the intersection points can be determined by solving the equation by a Newton iteration. If a) both conics are given implicitly (by an equation) a 2-dimensional Newton iteration b) one implicitly and the other parametrically given a 1-dimensional Newton iteration is necessary. See next section.
To show this, Gergonne considered lines through corresponding points of tangency on two of the given circles, e.g., the line defined by A 1 /A 2 and the line defined by B 1 /B 2. Let X 3 be a center of similitude for the two circles C 1 and C 2; then, A 1 /A 2 and B 1 /B 2 are pairs of antihomologous points, and their lines intersect at X 3. It ...
Consider, for example, the one-parameter family of tangent lines to the parabola y = x 2. These are given by the generating family F ( t ,( x , y )) = t 2 – 2 tx + y . The zero level set F ( t 0 ,( x , y )) = 0 gives the equation of the tangent line to the parabola at the point ( t 0 , t 0 2 ).